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\usepackage{amsmath, amsthm, amssymb, bm} % 数学公式与符号
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\title[数学论文写作]{数学论文写作的27个需要注意的地方和一个例子}
\author[DEK]{DEK ET AL}
%\institute[XX大学]{XX大学\quad 数学与统计学院\quad 数学与应用数学专业}
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%\date{2025年6月}

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\begin{enumerate}\itemsep1em
\item 数学论文写作的一些参考书
\item 数学论文写作的27个需要注意的地方(by Donald Knuth)
\item 一道习题的解答的四种写法(by Donald Knuth)
\end{enumerate}

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%\begin{frame}{\S1: Notes on Technical Writing - Stanford Library}
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%\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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%\begin{enumerate}\itemsep1em
%\item Stanford's library card catalog refers to more than 100 books about {\color{red}\bf technical writing}, including such titles as {\color{blue}\it The Art of Technical Writing}, {\color{blue}\it The Craft of Technical Writing}, {\color{blue}\it The Teaching of Technical Writing}. 
%
%\item There is even a journal devoted to the subject, the {\color{blue}\it IEEE Transactions on Professional Communication}, published since 1958. 
%
%\item The American Chemical Society, the American Institute of Physics, the American Mathematical Society, and the Mathematical Association of America have each published ``{\color{blue}manuals of style}.''
%
%\item The last of these, {\color{blue}\it Writing Mathematics Well} by Leonard Gillman, is one of the required texts for CS209.
%\end{enumerate}
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%\begin{frame}{\S1: Notes on Technical Writing - Stanford Library}
\begin{frame}{1.1. 与数学写作有关的一些中文参考书 }


\begin{enumerate}\itemsep1em

\item  汤涛, 丁玖. {\color{blue} 数学之英文写作}. 高等教育出版社, 2013年4月第1版. 
\item  韩茂安. {\color{blue} 数学研究与论文写作指导}. 科学出版社, 2018年7月第1版. 
\item  Nicholas J. Higham 著, 贾志刚, 常亮, 李建波译. {\color{blue} 数学论文写作}. 科学出版社, 2016年12月第1版. 
\item  William Strunk 著, 熊锡源译, 华研外语编. {\color{blue} 英语写作指南: 风格的要素}. 世界图书出版公司, 2020年4月第1版. 

\end{enumerate}

\end{frame}

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\begin{frame}{1.2. 与数学写作有关的一些英文参考书 }


\begin{enumerate}\itemsep1em
\item  Donald E. Knuth, Tracy L. Larrabee, and Paul M. Roberts. {\color{blue}\it Mathematical Writing}. MAA, 1989.
 
\item  Paul Halmos. {\color{blue}\it How to Write Mathematics}. AMS, 1973. 
\url{https://mathcomm.org/paul-halmos-on-writing-mathematics/}

\item  Nicholas J. Higham. {\color{blue}\it Handbook of Writing for the Mathematical Sciences}. Third Edition. SIAM, 2020. 

  \item  Steven G. Krantz. {\color{blue}\it A Primer of Mathematical Writing}. AMS, 2017. 

\item  William Strunk Jr.  {\color{blue}\it The Elements of Style}. Dover, 2005. 


\end{enumerate}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}{ References for CS209, 1987 }
%
%\begin{enumerate}
%\item The nicest little reference for a quick tutorial is {\color{blue}\it The Elements of Style}, by Strunk and White (Macmillan, 1979). 
%
%\item Everybody should read this 85-page book, which tells about English prose writing in general. 
%But it isn't a required text — it's merely recommended.
%
%\item The other required text for CS209 is {\color{blue}\it A Handbook for Scholars} by Mary-Claire van Leunen (Knopf, 1978). 
%
%\item This well-written book is a real pleasure to read, in spite of its unexciting title. 
%
%\item It tells about footnotes, references, quotations, and such things, done correctly instead of the old-fashioned ``op. cit.'' way.
%
%\end{enumerate}
%
%\end{frame}

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%\begin{frame}{ Other References}
%
%\begin{enumerate}
%\item {\color{red}\bf Mathematical writing} has certain peculiar problems that have rarely been discussed in the literature.
%
%\item Gillman's book refers to the three previous classics in the field: An article by Harley Flanders, {\it Amer. Math. Monthly}, 1971, pp. 1–10;
%another by R. P. Boas in the same journal, 1981, pp. 727–731.
%
%\item There's also a nice booklet called {\color{blue}\bf \it How to Write Mathematics}, published by the American Mathematical Society in 1973, especially the delightful essay by Paul R. Halmos on pp. 19–48.
%
%\item The following points are especially important, in your instructor's view.
%\end{enumerate}
%
%\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.1. }

\begin{itemize}\itemsep1em
\item[1.] {\color{red}\bf Symbols in different formulas} must be separated by words.

\vspace{0.5cm}

    \begin{itemize}\itemsep1em
    \item Bad: Consider $S_q, q<p$.
    \item Good: Consider $S_q$, where $q<p$.
    \end{itemize}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.2. }

\begin{itemize}\itemsep1em

\item[2.] {\color{red}\bf Don't start a sentence} with a symbol.

\vspace{0.5cm}

    \begin{itemize}\itemsep1em
    \item Bad: $x^n-a$ has $n$ distinct zeroes.
    \item Good: The polynomial $x^n-a$ has n distinct zeroes.
    \end{itemize}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.3. }

\begin{itemize}\itemsep1em
\item[3.] {\color{red}\bf Don't use the symbols} $\therefore,\Rightarrow,\forall,\exists$, replace them by the corresponding words.
(Except in works on logic, of course.)


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.4. }

\begin{itemize}\itemsep0.5em

\item[4.] {\color{red}\bf The statement just preceding a theorem, algorithm, etc.,} should be a complete sentence or should end with a colon.

    \begin{itemize}\itemsep0.5em
    \item Bad: 

    \vspace{0.3cm}

    \fbox{\begin{minipage}{10.5cm}
    {We now have the following\\
    Theorem. H(x) is continuous.}
    \end{minipage}}

    \vspace{0.3cm}
    
    %This is bad on three counts, including rule 2. It should be rewritten, for example, like this:
    \item Good: 

    \vspace{0.3cm}
    
    \fbox{\begin{minipage}{10.5cm}
    We can now prove the following result.\\
    Theorem. The function H(x) defined in (5) is continuous.
    \end{minipage}}    

    \vspace{0.3cm}
    
    \item Even better would be to replace the first sentence by a more suggestive motivation, tying the theorem up with the previous discussion.
    \end{itemize}

\end{itemize}

\end{frame}

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\begin{frame}{2.5. }

\begin{itemize}
\item[5.]  {\color{red}\bf The statement of a theorem} should usually be self-contained, not depending on the assumptions in the preceding text. %(See the restatement of the theorem in point 4.)


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.6. }

\begin{itemize}\itemsep1em

\item[6.] {\color{red}\bf The word ``we'' is often useful} to avoid passive voice.
    \begin{itemize}
    \item Bad: The following result can now be proved.
    \item Good: We can now prove the following result.
    \end{itemize}

\item But this use of ``we'' should be used in contexts where it means ``you and me together'', not a formal equivalent of ``I''. 

\item Think of a dialog between author and reader.

\item In most technical writing, ``I'' should be avoided, unless the author's persona is relevant.

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.7. }

\begin{itemize}\itemsep1em

\item[7.]  There is a definite {\color{red}\bf rhythm} in sentences. 
Read what you have written, and change the wording if it does not {\color{red}\bf flow smoothly}. 

\item  For example, in the text {\it Sorting and Searching} it was sometimes better to say ``merge patterns'' and sometimes better to say ``merging patterns''. 

\item  There are many ways to say ``therefore'', but often only one has the correct rhythm.

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.8. }

\begin{itemize}\itemsep1em

\item[8.] {\color{red}\bf Don't omit ``that''} when it helps the reader to parse the sentence.
\begin{itemize}
\item Bad: Assume A is a group.
\item Good: Assume that A is a group.
\end{itemize}

\item The words ``assume'' and ``suppose'' should usually be followed by ``that'' unless another ``that'' appears nearby.

\item  But never say ``We have that $x = y$,'' say ``We have $x = y$.'' 

\item  And avoid unnecessary padding ``because of the fact that'' unless you feel that the reader needs a moment to recuperate from a concentrated sequence of ideas.

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.9. }

\begin{itemize}

\item[9.] {\color{red}\bf Vary the sentence structure} and the choice of words, to avoid monotony. 
{\color{red}\bf But use parallelism} when parallel concepts are being discussed. 

\item For example (Strunk and White \#15), don't say this:

\begin{itemize}
\item Formerly, science was taught by the textbook method, while now the laboratory method is employed. Rather:

\item Formerly, science was taught by the textbook method; now it is taught by the laboratory method.
\end{itemize}

\item {\color{red}\bf Avoid words like ``this'' or ``also'' in consecutive sentences}; 
such words, as well as unusual or polysyllabic utterances, tend to stick in a reader's mind longer than other words, and good style will keep ``sticky'' words spaced well apart. 

%\item (For example, I’d better not say ``utterances'' any more in the rest of these notes.)

%\item 
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.10. }

\begin{itemize}
\item[10.] {\color{red}\bf Don't use the style of homework papers}, in which a sequence of formulas is merely listed.
Tie the concepts together with a running commentary.


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.11. }

\begin{itemize}\itemsep1em

\item[11.] {\color{red}\bf Try to state things twice}, in complementary ways, especially when giving a definition.
This reinforces the reader's understanding.

\item Examples, see \S2 below:\\
$N^n$ is defined twice, $A_n$ is described as ``nonincreasing'', $L(C,P)$ is characterized as the smallest subset of a certain type.

\item All variables must be defined, at least informally, when they are first introduced.

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.12. }

\begin{itemize}\itemsep1em

\item[12.] {\color{red}\bf Motivate the reader} for what follows. 
%\item In the example of \S2, Lemma 1 is motivated by the fact that its converse is true. Definition 1 is motivated only by decree; this is somewhat riskier.
Perhaps the most important principle of good writing is to keep the reader uppermost in mind: 

\vspace{0.2cm}

\fbox{\begin{minipage}{12cm}
What does the reader know so far? \\
What does the reader expect next and why?
\end{minipage}}

\vspace{0.2cm}

\item {\color{blue}\bf When describing the work of other people} it is sometimes safe to provide motivation by simply stating that it is ``interesting'' or ``remarkable''; but it is best to let the results speak for themselves or to give reasons why the things seem interesting or remarkable.

\item {\color{blue}\bf When describing your own work}, be humble and don't use superlatives of praise, either explicitly or implicitly, even if you are enthusiastic.

%\item 
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.13. }

\begin{itemize}

\item[13.] Many readers will skim over formulas on their first reading of your exposition. 
Therefore, your sentences should {\color{red}\bf flow smoothly} when all but the simplest formulas are replaced by ``blah'' or some other grunting noise.

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.14. }

\begin{itemize}\itemsep1em

\item[14.] {\color{red}\bf Don't use the same notation} for two different things. 
Conversely, {\color{red}\bf use consistent notation} for the same thing when it appears in several places. 

\item For example, don't say ``$A_j$ for $1 \le j \le n$'' in one place and ``$A_k$ for $1 \le k \le n$'' in another place unless there is a good reason.

\item It is often useful to choose names for indices so that $i$ varies from $1$ to $m$ and $j$ from $1$ to $n$, say, and to stick to consistent usage.

\item Typographic conventions (like lowercase letters for elements of sets and uppercase for sets) are also useful.

%\item 
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.15. }

\begin{itemize}\itemsep0.5em

\item[15.] {\color{red}\bf Don't get carried away by subscripts}, especially when dealing with a set that doesn't need to be indexed;
set element notation can be used to avoid subscripted subscripts.

\item For example, it is often troublesome to start out with a definition like ``Let $X = \{x_1,\cdots,x_n\}$'' if you're going to need subsets of $X$, since the subset will have to be defined as $\{x_{i_1},\cdots,x_{i_m}\}$, say.

\item Also you'll need to be speaking of elements $x_i$ and $x_j$ all the time.

\item Don't name the elements of $X$ unless necessary. 

\item Then you can refer to elements $x$ and $y$ of $X$ in your subsequent discussion, without needing subscripts; 
or you can refer to $x_1$ and $x_2$ as specified elements of $X$.

%\item 
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.16. }

\begin{itemize}\itemsep1em
\item[16.] {\color{red}\bf Display important formulas} on a line by themselves.

\item If you need to refer to some of these formulas from remote parts of the text, give reference numbers to all of the most important ones, even if they aren't referenced.

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.17. }

\begin{itemize}

\item[17.] Sentences should be readable from left to right {\color{red}\bf without ambiguity}. 

\vspace{0.5cm}

    Bad examples: 
    
    \begin{itemize}\itemsep1em
    \item ``Smith remarked in a paper about the scarcity of data.''
    \item ``In the theory of rings, groups and other algebraic structures are treated.''
    \end{itemize}

%\item 
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.18. }

\begin{itemize}
\item[18.] {\color{red}\bf Small numbers should be spelled out} when used as adjectives, but not when used as names (i.e., when talking about numbers as numbers).

\vspace{0.5cm}

\begin{itemize}\itemsep1em
\item Bad: The method requires 2 passes.
\item Good: Method 2 is illustrated in Fig. 1; it requires 17 passes. The count was increased by 2. The leftmost 2 in the sequence was changed to a 1.
\end{itemize}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.19. }

\begin{itemize}

\item[19.] Capitalize names like {\color{red}\bf Theorem 1}, Lemma 2, Algorithm 3, Method 4.

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.20. }

\begin{itemize}
\item[20.] Some handy maxims:

\vspace{0.5cm}

    \begin{itemize}\itemsep1em
    \item Watch out for prepositions that sentences end with.
    \item When dangling, consider your participles.
    \item About them sentence fragments.
    \item Make each pronoun agree with their antecedent.
    \item Don't use commas, which aren't necessary.
    \item Try to never split infinitives.
    \end{itemize}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.21. }

\begin{itemize}
\item[21.] Some words frequently misspelled by computer scientists:

\vspace{0.3cm}

\begin{tabular}{rcl}
implement & not & impliment \\
complement & not & compliment \\
occurrence & not & occurence \\
dependent & not & dependant \\
auxiliary & not & auxillary \\
feasible & not & feasable \\
preceding & not & preceeding \\
referring & not & refering \\
category & not & catagory \\
consistent & not & consistant \\
%PL/I & not & PL/1 \\
%descendant (noun) & not & descendent \\
%its (belonging to it) & not & it's (it is) \\
\end{tabular}

%The following words are no longer being hyphenated in current literature:
%nonnegative
%nonzero


%\item 
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.22. }

\begin{itemize}\itemsep1em

\item[22.] {\color{red}\bf Don't say ``which'' when ``that'' sounds better. }

\item The general rule nowadays is to use ``which'' only when it is preceded by a comma or by a preposition, or when it is used interrogatively.

\item Experiment to find out which is better, ``which'' or ``that'', and you'll understand this rule.
\begin{itemize}
\item Bad: Don't use commas which aren't necessary.
\item Better: Don't use commas that aren't necessary.
\end{itemize}

\item Another common error is to say ``less'' when the proper word is ``fewer''.


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.23. }

\begin{itemize}
\item[23.] In the example at the bottom of \S2 below, note that the text preceding displayed equations (1) and (2) does not use any special punctuation. Many people would have written

\vspace{0.2cm}

\begin{center}
\fbox{\begin{minipage}{12cm}
... of ``nonincreasing'' {\color{red}\bf vectors:}
\begin{eqnarray}
A_n = \{(a_1,\cdots,a_n) \in N^n \mid a_1 \ge \cdots \ge a_n \}.
\end{eqnarray}

If $C$ and $P$ are subsets of $N^n$ , {\color{red}\bf let:}
\begin{eqnarray*}
L(C,P) = \cdots
\end{eqnarray*}

\end{minipage}}
\end{center}

\vspace{0.2cm}

and those colons are wrong.

%\item 
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.24.a.  }

\begin{itemize}\itemsep1em

\item[24.a.] {\color{red}\bf The opening paragraph} should be your best paragraph, {\color{red}\bf and its first sentence} should be your best sentence.

\item If a paper starts badly, the reader will wince and be resigned to a difficult job of fighting with your prose. 

\item Conversely, if the beginning flows smoothly, the reader will be hooked and won't notice occasional lapses in the later parts.

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.24.b.  }

\begin{itemize}\itemsep1em

\item[24.b.] Probably the worst way to start is with a sentence of the form 

{\color{red}\bf ``An $x$ is $y$.'' } 

\item  For example,

\begin{itemize}
\item Bad: An important method for internal sorting is quicksort.
\item Good: Quicksort is an important method for internal sorting, because ...
\end{itemize}

\begin{itemize}
\item Bad: A commonly used data structure is the priority queue.
\item Good: Priority queues are significant components of the data structures needed for many different applications.
\end{itemize}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.25. }

\begin{itemize}\itemsep1em

\item[25.] The normal style rules for English say that {\color{red}\bf commas and periods} should be placed inside quotation marks, but other punctuation (like colons, semicolons, question marks, exclamation marks) stay outside the quotation marks unless they are part of the quotation. 

\item It is generally best to go along with this illogical convention about commas and periods, because it is so well established, except when you are using quotation marks to describe some text as a specific string of symbols. 

\item For example,
\begin{itemize}
\item Good: Always end your program with the word ``end''.
\item Bad: This is bad, (although intentionally so.)
\end{itemize}

%\item On the other hand, punctuation should always be strictly logical with respect to parentheses and brackets. 
%
%\item Put a period inside parentheses if and only if the sentence ending with that period is entirely within the parentheses.
%
%\item The punctuation within parentheses should be correct, independently of the outside context, and the punctuation outside the parentheses should be correct if the parenthesized statement would be removed.

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.26.a. }

\begin{itemize}\itemsep1em

\item[26a.] {\color{red}\bf Resist the temptation} to use long strings of nouns as adjectives: consider the packet switched data communication network protocol problem.

%\item In general, don't use jargon unnecessarily. 

\item Even specialists in a field get more pleasure from papers that use a nonspecialist's vocabulary.


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.26.b. }

\begin{itemize}

\item[26b.] Bad: 

\vspace{0.2cm}

\begin{center}
\fbox{\begin{minipage}{12.5cm}
If $L^+(P,N_0)$ is the set of functions $f:P \to N_0$ with the property that
\begin{eqnarray*}
\underset{n_0\in N_0}{\exists}\,\,\, \underset{p\in P}{\forall}\,\,\, p\ge n_0 \Rightarrow f(p)=0
\end{eqnarray*}
then there exists a bijection $N_1 \to L^+(P,N_0)$ such that if $n\to f$ then
\begin{eqnarray*}
n=\underset{p\in P}{\Pi}\,\,\, p\,^{f (p)}.
\end{eqnarray*}
Here $P$ is the prime numbers and $N_1 = N_0\sim \{0\}$.
\end{minipage}}
\end{center}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.26.c. }

\begin{itemize}
\item[26c.] Better: 

\vspace{0.2cm}

\begin{center}
\fbox{\begin{minipage}{12.5cm}
According to the `fundamental theorem of arithmetic' (proved in ex. 1.2.4–21), each positive integer $u$ can be expressed in the form
\begin{eqnarray*}
u=2^{u_2}3^{u_3}5^{u_5}7^{u_7}11^{u_{11}}\cdots = \underset{p\,\, \textrm{prime}}{\Pi} p^{u_p}
\end{eqnarray*}
where the exponents $u_2,u_3,\cdots$ are uniquely determined nonnegative integers, and where all but a finite number of the exponents are zero.
\end{minipage}}
\end{center}

%\item [The first quotation is from Carl Linderholm's neat satirical book {\it Mathematics Made Difficult}; the second is from D. Knuth's {\it Seminumerical Algorithms}, Section 4.5.2.]

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{2.27. }

\begin{itemize}\itemsep1em

\item[27.] When in doubt, read {\color{blue}\it The Art of Computer Programming} for outstanding examples of good style.

\item That was a joke. Humor is best used in technical writing when readers can {\color{red}\bf understand the joke} only when they also understand a technical point that is being made. Here is another example from Linderholm:

\item `` $\cdots \emptyset D = \emptyset$ and $N\emptyset = N$, which we may express by saying that $\emptyset$ is absorbing on the left and neutral on the right, like British toilet paper.''

\item Try to restrict yourself to jokes that will not seem silly on second or third reading. And don't overuse exclamation points!


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.1. An Exercise on Technical Writing}

\begin{itemize}\itemsep1em

\item In the following excerpt from a term paper, 

\vspace{0.3cm}

    \begin{itemize}\itemsep1em
    \item $N$ denotes the nonnegative integers, $N^n$ denotes the set of $n$-tuples of nonnegative integers, and $A_n = \{(a_1,\cdots,a_n) \in N^n \mid a_1 \ge \cdots \ge a_n \}$.
    \item If $C,P \subset N^n$, then $L(C,P)$ is defined to be $\{c+p_1+\cdots+p_m \mid c\in C, m \ge 0,\textrm{ and }p_j\in P
    \textrm{ for } 1 \le j \le m\}$. 
    \item We want to prove that $L(C,P) \subseteq A_n$ implies $C,P \subseteq A_n$.
    \end{itemize}

\item The following proof, directly quoted from a sophomore term paper, is {\color{red}\bf mathematically correct} (except for a minor slip) but {\color{red}\bf stylistically atrocious}:

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.2. First Solution }

{\small

$L(C,P) \subset A_n$

$C\subset L \Rightarrow C \subset A_n$

Spse $p \in P, p \notin An \Rightarrow p_i < p_j$ for $i < j$

$c + p \in L \subset A_n$

$\therefore c_i + p_i \ge c_j + p_j$ but $c_i \ge c_j \ge 0, p_j \ge p_i$ $\therefore (c_i - c_j) \ge (p_j - p_i)$

but $\exists$ a constant $k \ni c + kp \notin A_n$

let $k = (c_i - c_j)+1$ \hspace{0.5cm} $c + kp \in L \subset A_n$

$\therefore c_i + kp_i \ge c_j + kp_j \Rightarrow (c_i - c_j) \ge k(p_j - p_i)$

$\Rightarrow k-1 \ge k\cdot m$ \hspace{0.5cm} $k,m \ge 1$ \hspace{0.5cm} Contradiction

$\therefore p \in A_n$

$\therefore L(C,P) \subset A_n \Rightarrow C,P \subset A_n$ and the

lemma is true.

}

\end{frame}

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\begin{frame}{3.3. Second Solution - Part 1}

\begin{itemize}
\item A possible way to improve the quality of the writing:

\end{itemize}


\begin{center}
\fbox{\begin{minipage}{13.5cm}
Let $N$ denote the set of nonnegative integers, and let
\begin{eqnarray*}
N^n = \{(b_1,\cdots,b_n) \mid b_i\in N \textrm{ for } 1 \le i \le n\}
\end{eqnarray*}
be the set of $n$-dimensional vectors with nonnegative integer components.

We shall be especially interested in the subset of ``nonincreasing'' vectors,
\begin{eqnarray}
A_n = \{(a_1,\cdots,a_n) \in N^n \mid a_1 \ge \cdots \ge a_n \}
\end{eqnarray}

\end{minipage}}
\end{center}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.4. Second Solution - Part 2 }

\begin{center}
\fbox{\begin{minipage}{13.5cm}

If $C$ and $P$ are subsets of $N^n$, let
\begin{eqnarray}
L(C,P) = \{c + p_1 + \cdots + p_m \mid c\in C,m\ge 0, \textrm{ and } p_j \in P \textrm{ for } 1 \le j \le m\}
\label{LCP}
\end{eqnarray}
be the smallest subset of $N^n$ that contains $C$ and is closed under the addition of elements of $P$.

\end{minipage}}
\end{center}


\begin{center}
\fbox{\begin{minipage}{13.5cm}

Since $A_n$ is closed under addition, $L(C,P)$ will be a subset of $A_n$ whenever $C$ and $P$ are both contained in $A_n$. We can also prove the converse of this statement.

\end{minipage}}
\end{center}


\begin{center}
\fbox{\begin{minipage}{13.5cm}

Lemma 1. If $L(C,P) \subseteq A_n$ and $C\neq\emptyset$, then $C\subseteq A_n$ and $P \subseteq A_n$.

Proof. (Now it's your turn to write it up beautifully.)

\end{minipage}}
\end{center}

%\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.5. Third Solution - Part 1  }

\begin{itemize}\itemsep1em

\item Here is one way to complete the exercise in the previous section.

\item (But please try to work it yourself before reading this.)

\item Note that a few clauses have been inserted to help keep the reader synchronized with the current goals and subgoals and strategies of the proof.

\item Furthermore the notation $(b_1,\cdots,b_n)$ is used instead of $(p_1,\cdots,p_n)$, in the second paragraph below, to avoid confusion with formula (\ref{LCP}).

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.6. Third Solution - Part 2 }

\begin{center}
\fbox{\begin{minipage}{14.5cm}

{\footnotesize
{\color{blue}Proof.} Assume that $L(C,P) \subseteq A_n$. Since $C$ is always contained in $L(C,P)$, we must have $C \subseteq A_n$; therefore only the condition $P\subseteq A_n$ needs to be verified.
If $P$ is not contained in $A_n$, there must be a vector $(b_1,\cdots,b_n) \in P$ such that $b_i < b_j$ for some $i<j$. We want to show that this leads to a contradiction.
Since the set $C$ is nonempty, it contains some element $(c_1,\cdots,c_n)$.
We know that the components of this vector satisfy $c_1 \ge \cdots \ge c_n$, because $C \subseteq A_n$.

Now $(c_1,\cdots,c_n) + k(b_1,\cdots,b_n)$ is an element of $L(C,P)$ for all $k\ge 0$, and by hypothesis it must therefore be an element of $A_n$. But if we take $k = c_i - c_j + 1$, we have $k \ge 1$ and
\begin{eqnarray*}
c_i+kb_i \ge c_j+kb_j,
\end{eqnarray*}

hence
\begin{eqnarray}
c_i - c_j \ge k(b_j-b_i).
\end{eqnarray}

This is impossible, since $c_i-c_j=k-1$ is less than $k$, yet $b_j-b_i\ge 1$.
It follows that $(b_1,\cdots,b_n)$ must be an element of $A_n$. \hfill $\blacksquare$
}

\end{minipage}}
\end{center}

%Note that the hypothesis $C \neq \emptyset$ is necessary in Lemma 1, for if $C$ is empty the set $L(C,P)$ is also empty regardless of $P$. [This was the ``minor slip.'']

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.7. Fourth Solution - Part 1 }

\begin{itemize}\itemsep1em

\item BUT ... don't always use the first idea you think of. 

\item The proof above actually commits another sin against mathematical exposition, namely the unnecessary use of proof by contradiction. 

\item It would have been better to use a direct proof.

\item This form of the proof has other virtues too: It doesn't assume that the $b_i$'s are integer-valued, and it doesn't require stating that $c_1\ge \cdots \ge c_n$.
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.8. Fourth Solution - Part 2  }

\begin{center}
\fbox{\begin{minipage}{13.5cm}

{\color{blue}Proof.}
Let $(b_1,\cdots,b_n)$ be an arbitrary element of $P$, and let $i$ and $j$ be fixed subscripts with $i<j$;
we wish to prove that $b_i\ge b_j$. Since $C$ is nonempty, it contains some element $(c_1,\cdots,c_n)$.
Now the vector $(c_1,\cdots,c_n) + k(b_1,\cdots,b_n)$ is an element of $L(C,P)$ for all $k \ge 0$, and by hypothesis it must therefore be an element of $A_n$. But this means that $c_i +kb_i \ge c_j +kb_j$, i.e.,
\begin{equation}%\tag{3}
c_i-c_j \ge k(b_j-b_i),
\end{equation}
for arbitrarily large $k$. Consequently $b_j-b_i$ must be zero or negative. We have proved that $b_j-b_i\le 0$ for all $i<j$, so the vector $(b_1,\cdots,b_n)$ must be an element of $An$. \hfill $\blacksquare$

\end{minipage}}
\end{center}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.9. The End Note}

\vspace{2cm}

\begin{center}
{\Large \color{blue} Practice Makes Progress. }
\end{center}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

